Vector help please

1. The problem statement, all variables and given/known data
Three units vectors a, b, and c have property that the angle between any two is a fixed angle [tex]\theta[/tex]

(i) find in terms of [tex]\theta[/tex] the length of the vector v = a + b + c
(ii) find the largest possible value of [tex]\theta[/tex]
(iii) find the cosine of the angle [tex]\beta[/tex] between a and v

2. Relevant equations
unit vector = vector with length 1unit

magnitude of vector = [tex]\sqrt{x^2+y^2+z^2}[/tex]

[tex]\cos \theta = \frac{r_1\cdot r_2}{|r_1||r_2|}[/tex]

3. The attempt at a solution
(i) I think I get it right. The answer is [tex]\sqrt{3+6\cos \theta}[/tex]

(ii) I don't know how to do this. I think [tex]\theta < 90^o[/tex] , but I can't find the exact value

first though, the way to visualise this is to consider all the vectors pointing in the same direction, theta = 0. this is where |v| = 3

as the angle is increased, imagine the vectors spreading something like a flower opening, keeping the same angle between each, with |v| decreasing. The maximum angle occurs when they are all in a plane, theta = 120, and |v| = 0. Agreeing with the first range of your solution.

I also think you only need to consider upto 120 (solutions for 120<theta<= 180 do not exist, and above 180 you can just measure the angle the other way)